Find the greatest common factor of $36$ and $63$.
Solution: The greatest common factor (GCF) is the largest number that is a factor of both $36$ and $63$. In order to find the GCF, we can factor each number completely as a product of prime numbers: $ \begin{aligned}36 &=2\cdot2\cdot3\cdot3\\\\\\\\ 63&=3\cdot3\cdot7 \end{aligned}$ Now, let's find the factors that are common to each number: $ \begin{aligned}36 &=2\cdot2\cdot3\cdot3\\\\\\\\ 63&=3\cdot3\cdot7 \end{aligned}$ Each number shares the factors ${3}$ and ${3}$, so the GCF is $3\cdot3=9$. The greatest common factor of $36$ and $63$ is $9$.